The Gröbner fan of an An-module

Abstract

Let I be a non-zero left ideal of the Weyl algebra An of order n over a field k and let L:R2n→R be a linear form defined by L(α,β)=∑i=1neiαi+∑i=1nfiβi. If ei+fi≥0, then L defines a filtration F•L on An. Let grL(I) be the graded ideal associated with the filtration induced by F•L on I. Let finally U denote the set of all linear form L for which ei+fi≥0 for all 1≤i≤n. The aim of this paper is to study, by using the theory of Gröbner bases, the stability of grL(I) when L varies in U. In a previous paper, we obtained finiteness results for some particular linear forms (used in order to study the regularity of a D-module along a smooth hypersurface). Here we generalize these results by adapting the theory of Gröbner fan of Mora-Robbiano to the D-module case. Our main tool is the homogenization technique initiated in our previous paper, and recently clarified in a work by F. Castro-Jiménez and L. Narváez-Macarro.